The Bloch–Kato Conjecture for the Riemann Zeta Function

doi:10.1017/CBO9781316163757

There are still many arithmetic mysteries surrounding the values of the Riemann zeta function at the odd positive integers greater than one. For example, the matter of their irrationality, let alone transcendence, remains largely unknown. However, by extending ideas of Garland, Borel proved that these values are related to the higher K-theory of the ring of integers. Shortly afterwards, Bloch and Kato proposed a Tamagawa number-type conjecture for these values, and showed that it would follow from a result in motivic cohomology which was unknown at the time. This vital result from motivic cohomology was subsequently proven by Huber, Kings, and Wildeshaus. Bringing together key results from K-theory, motivic cohomology, and Iwasawa theory, this book is the first to give a complete proof, accessible to graduate students, of the Bloch–Kato conjecture for odd positive integers. It includes a new account of the results from motivic cohomology by Huber and Kings.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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List of contributors
pp vii-vii
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Preface
pp viii-x
- By A. Raghuram, Indian Institute of Science Education and Research
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1 - Special Values of the Riemann Zeta Function: Some Results and Conjectures
pp 1-21
- By A. Raghuram, Indian Institute of Science Education and Research, Pune, India
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2 - K-theoretic Background
pp 22-44
- By R. Sujatha, University of British Columbia, Vancouver, Canada
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3 - Values of the Riemann Zeta Function at the Odd Positive Integers and Iwasawa Theory
pp 45-64
- By John Coates, University of Cambridge, UK
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4 - Explicit Reciprocity Law of Bloch–Kato and Exponential Maps
pp 65-96
- By Anupam Saikia, Indian Institute of Technology Guwahati, India
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5 - The Norm Residue Theorem and the Quillen-Lichtenbaum Conjecture
pp 97-120
- By Manfred Kolster, McMaster University, Hamilton, Canada
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6 - Regulators and Zeta-functions
pp 121-129
- By Stephen Lichtenbaum, Brown University, Providence, USA
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7 - Soulé's Theorem
pp 130-139
- By Stephen Lichtenbaum, Brown University, Providence, USA
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8 - Soulé's Regulator Map
pp 140-153
- By Ralph Greenberg, University of Washington, Seattle, USA
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9 - On the Determinantal Approach to the Tamagawa Number Conjecture
pp 154-192
- By T. Nguyen Quang Do, University of Franche-Comté, France
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10 - Motivic Polylogarithm and Related Classes
pp 193-209
- By Don Blasius, University of California at Los Angeles, USA
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11 - The Comparison Theorem for the Soulé–Deligne Classes
pp 210-238
- By Annette Huber, Albert-Ludwigs-Universität Freiburg, Germany
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12 - Eisenstein Classes, Elliptic Soulé Elements and the ℓ-Adic Elliptic Polylogarithm
pp 239-296
- By Guido Kings, University of Regensburg, Germany
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13 - Postscript
pp 297-305
- By R. Sujatha, University of British Columbia, Vancouver, Canada
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The Bloch–Kato Conjecture for the Riemann Zeta Function

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

List of contributors
pp vii-vii

Preface
pp viii-x

1 - Special Values of the Riemann Zeta Function: Some Results and Conjectures
pp 1-21

2 - K-theoretic Background
pp 22-44

3 - Values of the Riemann Zeta Function at the Odd Positive Integers and Iwasawa Theory
pp 45-64

4 - Explicit Reciprocity Law of Bloch–Kato and Exponential Maps
pp 65-96

5 - The Norm Residue Theorem and the Quillen-Lichtenbaum Conjecture
pp 97-120

6 - Regulators and Zeta-functions
pp 121-129

7 - Soulé's Theorem
pp 130-139

8 - Soulé's Regulator Map
pp 140-153

9 - On the Determinantal Approach to the Tamagawa Number Conjecture
pp 154-192

10 - Motivic Polylogarithm and Related Classes
pp 193-209

11 - The Comparison Theorem for the Soulé–Deligne Classes
pp 210-238

12 - Eisenstein Classes, Elliptic Soulé Elements and the ℓ-Adic Elliptic Polylogarithm
pp 239-296

13 - Postscript
pp 297-305

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The Bloch–Kato Conjecture for the Riemann Zeta Function

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